[Verse 1] Every wave that breaks the silence, every heartbeat in the night Can be carved into frequencies, pure harmonics burning bright Take your signal, complex and wild, decompose what seems obscure Into sines and cosines dancing, each component clean and pure [Chorus] Project onto orthonormal basis, find the coefficients hiding there A-sub-n equals inner product, f with e-sub-n in the air Infinite dimensions waiting, function space like vectors grown Fourier transforms the chaos into frequencies we've always known [Verse 2] Orthogonality's the secret, why this magic reconstruction works Sines and cosines never intersect, their inner product always shirks From the integral of zero comes the power to decode Every reasonable periodic into its harmonic code [Chorus] Project onto orthonormal basis, find the coefficients hiding there A-sub-n equals inner product, f with e-sub-n in the air Infinite dimensions waiting, function space like vectors grown Fourier transforms the chaos into frequencies we've always known [Bridge] Parseval whispers energy preserved Total power stays conserved Integral of f-squared equals Sum of coefficients revealed L-two convergence, pointwise, uniform Convergence theorems keep us warm [Verse 3] This is spectral theory's prototype, eigenvalues decomposed Representation theory echoes, group actions now exposed Wavelets localize the frequency, but the principle remains Complex objects yield to simple parts, orthogonal refrains [Chorus] Project onto orthonormal basis, find the coefficients hiding there A-sub-n equals inner product, f with e-sub-n in the air Infinite dimensions waiting, function space like vectors grown Fourier transforms the chaos into frequencies we've always known [Outro] From applied mathematics flowing This idea keeps on growing Decompose into pieces orthogonal The most productive call
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